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The Princeton Companion to Mathematics?
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Hey! Fellow sellout here.
I don't want to gloss over the fact that it's a massive lift, but I'm about 100 pages into the Princeton Companion (https://smile.amazon.com/dp/0691118809/ref=cm_sw_r_cp_apa_glt_fabc_BMQ2QH2XA4C6B34M130J) and I personally love it. It's extremely well cross referenced so you do not need to read it cover-to-cover.
I love it because it gives just enough depth for you to get an intuitive flavor of different topics and areas. It tells me just enough, without pulling punches, for me to tell if a topic is something I want to look more into and in a serious way.
I had a similar request to yours, except I wanted to go beyond Calculus to get a broad survey of mathematical topics, using a ground up approach. The Princeton Companion to Mathematics is exceptional, I can't recommend it enough! It covers all the topics you wish your mathematics teachers had instilled in you, all within a comprehensive & comprehensible form. It has been years since I studied math. I've long since forgotten a majority of what I was taught but, I can still easily progress in this book and I feel like I finally understand many of the ideas that were impenetrable before.
I'm not alone in my positive review. You'll note that people have been heaping praise onto this volume on Amazon and in more formal book reviews as well.
I second the Companion. It's $63 new on Amazon, and reading it has given me a much broader understanding of modern math.
I used it for college calc and linear algebra, but if you are looking for past that then no, they probably don't have much.
One book I'm unwilling to part with is the princeton companion to mathematics. It's an incredible book with an entry by well known mathematicians in their field, writing on every major field of math, in a way that is accessible to people with a college level math background. It would be a really good starting point to find some sub-topics you were interested in.
Something to find from the public library maybe?
I recommend that you buy the book Princeton Companion to Mathematics. Tim Gowers, the editor (of this 1000+ page tome), describes the book as a long introduction to pure mathematics. If you're interested in grad school in math, this could be a great way to see more of what's out there. It has 26 "long articles" on the branches of math, such as 12 pages on Differential Topology, as well as 99 "short articles" (ranging in length from 1/2 page to 5 pages), including one on "Lie Theory", and much more. The authors of the article were chosen equally for the fame as mathematicians and for the skill at explaining concepts clearly and simply.
I know this doesn't directly answer the question you asked, but I would recommend looking at The Princeton Companion to Mathematics because it's aimed at people with an undergrad understanding of math and covers emerging research. It might help you get a better idea of where you want to take your research in the future. I wish it had existed when I was an undergrad.
Have you considered buying him a book on mathematics? I know they can be expensive but there are some interesting ones out there.
For example, the bible of the mathematics.
I can't believe this hasn't been suggested yet: The Princeton Companion to Mathematics
Actually, it's really only if you're interested in pure mathematics. If you're more interested in the applied side, look elsewhere.
The Princeton Companion to Mathematics has been a nice coffee table book.
Check out The Princeton Companion to Mathematics.
https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
> I think the point is that pseudo random attempts to generate results that “appear” random. That is, you can’t reliably predict the next value with certainty. It’s pseudo because, if you knew the prior conditions, you /could/ predict the next value with certainty.
This is why I said it can be difficult to define randomness. Strictly speaking, randomness isn't a property of sequences or numbers, or anything mathematical; it's a property of physical processes. Suppose we are given some sequence of numbers and are asked if it is random: if we don't know where it came from, then we just can't say. For instance, the sequence of numbers "1, 2, 3, 4" might have been produced by chance by some (as far as we can tell) purely random physical process, or it might have been constructed according to the rule "obtain an item in the sequence by adding 1 to the previous item", or it might have been gotten by looking at the 13807^th and subsequent digits of pi. Without knowing how it was produced, we can't tell. (It might seem unlikely that it''s random, but it's certainly possible, especially for such a short sequence.)
One textbook definition (not especially precise, but it's a good example of the sort of "working definition" that many fields of science use), is:
> scientists use chance, or randomness, to mean that when physical causes can result in any of several outcomes, we cannot predict what the outcome will be in any particular case. (Futuyma, D.J., 2005, Evolution, cited in SEP "Chance vs Randomness")
One reason this definition is not precise is that it's not clear whether something is random when it just so happens that scientists at a particular time are unable to predict the outcome, or whether something stronger is required - that it be impossible in principle to predict the outcome. It's also not really clear who the "we" is - can something be random for one person but not another?
In most fields of science, scientists just aren't especially interested in narrowing down the meaning of "randomness" any precisely than the way Futuyma gives it - as long as it works, they're happy.
If you're interested in delving deeper into how randomness and pseudo-randomness are defined, then I can recommend two good places to look: the <em>Princeton Companion to Mathematics</em> has a good discussion of pseudo-random number generators in section IV.20.6, and the "Chance versus randomness" article from the Stanford Encyclopedia of Philosophy (which I already linked to) has a good discussion of conceptual issues related to randomness. (I don't think the Wikipedia article on randomness is especially illuminating, but I guess you can look there also.)
> I forget where I read this, but wasn’t it Steve Jobs that said that a randomized process may result in two songs from the same album being played back to back, so their algorithm was modified and explicitly defined to /never have that happen/ it is, then, “less random”—whatever that means—but it appears more “correct” to humans because we fallaciously expect low-chance events to never happen. We have a psychological bias as to what random “ought to be”
Yes, other commenters have already noted this point. I have no idea off-hand who said it, though I suspect Steve Jobs probably wasn't the first person to say it or notice it. As you seem to imply, it has more to do with human psychology than randomness - humans just don't have a very good intuition for when events aren't following a pattern. ,
> My solution to this quandary?
Well, there is no "quandary". One way of generating a "shuffled playlist" is to select each song (without replacement) with equal probability; it just turns out that's not how most people want a shuffled playlist to behave.
> Which is necessarily true except for some subatomic particles…
To the best of our knowledge, quantum events are unpredictable; but I don't know that that's the same as saying that it's "necessarily true" as a matter of logic, if that's what you're implying. I'd suggest that S.E.P. article, again, for further discussion of the issue.
> which I’d say can still be determined in one dimension or another,
I'm afraid I have no idea what this means.
I hope my response has been useful.
I think there's like a Princeton's math companion that many here have suggested it in the past, but I have no personal exp with it:
https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
Also you might like this series:
https://www.amazon.com/-/es/gp/aw/d/069111384X/ref=dbs_a_w_dp_069111384x
https://www.amazon.com/-/es/gp/aw/d/0691113858/ref=dbs_a_w_dp_0691113858
https://www.amazon.com/-/es/gp/aw/d/0691113866/ref=dbs_a_w_dp_0691113866
Haven't studied them either, but let's say my intuition tells me they are worth your time.
Of course. More just generalizable/widely applicable fundamentals.
Example: https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
Princeton companion to mathematics https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809?
Rather than give descriptions of each of the fields you named, I'll just mention that you might want to peruse The Princeton Companion to Mathematics.
https://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809
Expensive, but worth it.
Your enthusiasm is great. Good for you.
You are doing well so far. As you move along, enjoy your classes. Take associated subjects also, like Physics, Econ, and CS. Lots of math is informed and inspired by applications.
If you want to get a sense of what you are moving toward you could leaf through The Princeton Companion to Mathematics. It is a very exciting book; of course much of it would be beyond a person without any background but it allows a person some idea of the work being done.
When you are looking at colleges, ask about the math program. Almost any school with have a math degree, but ask about what the program features or emphasizes. Feel free to email the chair of the department of those schools you are interested in (I was a chair and I was delighted to get emails like that).
When you are in college, the advice to take undergrad research is sound. Get to know your profs, and see if you can hook into something, like a problem seminar. Any reasonable person love to find a student who wants to know more, who thinks that what they are saying is neat. So don't be shy about asking about anything in the class.
Best thing you can do is read your little heart out. Find a copy (electronic or library) of something like the Princeton Companion and browse it over the course of a few weeks/months and pick out a few fields that particularly interest you. Then hit up How To Become A Pure Mathematician and start on your reading.
Eventually - and by this I mean "in a year or two" - you want to be able to email the prof in your dept who shares your interest and say "Hey, I've read the foundational texts on xyz, what would you recommend next?" and from there develop a relationship that'll hopefully lead to some undergrad "research" and a glowing letter of recommendation in your final year.
The other, equally important thing is to be a likable, sociable person. Unless you're some kind of wunderkind, collaboration is the name of the game and it gives you a huge advantage over the smelly nerd that no-one really wants around.
e: also lol undergrad pure maths research hahahahaha. if you can read a contemporary research paper in most pure maths subfields by senior year, you're ahead of the game.
This will really pique his interest in math, even if he won't understand much(any) of it yet.
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